高中数学第一章三角函数13三角函数的诱导公式同步优化训练新人教A版必修4

1.3 三角函数的诱导公式

5分钟训练(预习类训练，可用于课前)

1.(高考湖南卷，文2)tan600°的值是( )

A.word/media/image3_1.png B.word/media/image4_1.png C.word/media/image5_1.png D.word/media/image6_1.png

解析：tan600°=tan(360°+240°)=tan240°=tan(180°+60°)=tan60°=word/media/image7_1.png.

答案：D

2.下列各式中成立的是( )

A.sin(-20°)+sin200°=0

B.sin370°-sin(-190°)=0

C.cos(3π+word/media/image8_1.png)=cos(-word/media/image8_1.png)

D.cosword/media/image9_1.png=cos(word/media/image10_1.png)

解析：sin(-20°)+sin200°=-sin20°+sin(180°+20°)=-2sin20°;

sin370°-sin(-190°)=sin10°+sin(180°+10°)=sin10°-sin10°=0;

cos(3π+word/media/image8_1.png)=cos(π+word/media/image8_1.png)=-cosword/media/image8_1.png≠cos(-word/media/image8_1.png)=cosword/media/image8_1.png;

cosword/media/image9_1.png=cos(4π+word/media/image11_1.png)=cosword/media/image11_1.png≠cos(word/media/image12_1.png)=cos(4π-word/media/image13_1.png)=cos(π-word/media/image11_1.png)=-cosword/media/image11_1.png.

答案：B

3.已知f(x)=word/media/image14_1.png,若α∈(word/media/image15_1.png,π)，则f(cosα)+f(-cosα)可化为_________________.

解析：f(cosα)+f(-cosα)

=word/media/image16_1.png.

答案：word/media/image17_1.png

4.求下列三角函数值：

(1)sinword/media/image18_1.png;(2)cosword/media/image19_1.png;(3)tan(word/media/image20_1.png)；(4)sin(-765°).

解：(1)sinword/media/image18_1.png=sin(2π+word/media/image21_1.png)=sinword/media/image21_1.png=word/media/image22_1.png.

(2)cosword/media/image19_1.png=cos(4π+word/media/image8_1.png)=cosword/media/image8_1.png=word/media/image23_1.png.

(3)tan(word/media/image20_1.png)=tan(-4π+word/media/image11_1.png)=tanword/media/image11_1.png=word/media/image24_1.png.

(4)sin(-765°)=sin［360°×(-2)-45°］=sin(-45°)=-sin45°=word/media/image25_1.png.

10分钟训练(强化类训练，可用于课中)

1.tan300°+sin450°的值是( )

A.word/media/image26_1.png B.word/media/image27_1.png C.word/media/image28_1.png D.word/media/image29_1.png

解析：tan300°+sin450°=word/media/image30_1.png+sin(360°+90°)=-tan60°+sin90°=word/media/image31_1.png.

答案：B

2.word/media/image32_1.png化简的结果是( )

A.sin3-cos3 B.cos3-sin3 C.±(sin3-cos3) D.以上都不对

解析：word/media/image32_1.png

=word/media/image33_1.png=｜cos3-sin3｜.

word/media/image15_1.png＜3＜π,∴sin3＞0＞cos3.

∴原式=sin3-cos3.

答案：A

3.如果α+β=180°，那么下列等式中成立的是( )

A.cosα=cosβ B.cosα=-cosβ

C.sinα=-sinβ D.以上都不对

解析：cosα=cos(180°-β)=-cosβ.

答案：B

4.已知cos(-100°)=a，求tan80°.

解：cos(-100°)=cos100°=cos(180°-80°)=-cos80°=a.

∴cos80°=-a,sin80°=word/media/image34_1.png.

∴tan80°=word/media/image35_1.png.

5.设f(θ)=word/media/image36_1.png，求f(word/media/image21_1.png)的值.

解：f(θ)=word/media/image37_1.png

=word/media/image38_1.png

=word/media/image39_1.png

=cosθ-1.

∴f(word/media/image21_1.png)=cosword/media/image21_1.png-1=word/media/image40_1.png-1=word/media/image41_1.png.

30分钟训练(巩固类训练，可用于课后)

1.如果|cosx|=cos(x+π)，则x的取值集合是( )

A.-word/media/image15_1.png+2kπ≤x≤word/media/image15_1.png+2kπ

B.-word/media/image42_1.png+2kπ≤x≤word/media/image43_1.png+2kπ

C.word/media/image15_1.png+2kπ≤x≤word/media/image43_1.png+2kπ

D.(2k+1)π≤x≤2(k+1)π(以上k∈Z)

解析：由｜cosx｜=-cosx，可知cosx≤0,所以 word/media/image15_1.png+2kπ≤x≤word/media/image43_1.png+2kπ,k∈Z.

答案：C

2.sin(word/media/image10_1.png)的值是( )

A.word/media/image40_1.png B.word/media/image41_1.png C.word/media/image22_1.png D.word/media/image44_1.png

解析：sin(word/media/image10_1.png)=sin(-2×2π+word/media/image13_1.png)=sinword/media/image13_1.png=sin(π-word/media/image11_1.png)=sinword/media/image11_1.png=word/media/image40_1.png.

答案：A

3.下列三角函数，其中函数值与sinword/media/image21_1.png的值相同的是( )

①sin(nπ+word/media/image45_1.png) ②cos(2nπ+word/media/image11_1.png) ③sin(2nπ+word/media/image21_1.png) ④cos［(2n+1)π-word/media/image11_1.png］ ⑤sin［(2n+1)π-word/media/image21_1.png］(以上n∈Z)

A.①② B.①③④ C.②③⑤ D.①③⑤

解析：②cos(2nπ+word/media/image11_1.png)=cosword/media/image11_1.png=word/media/image22_1.png=sinword/media/image21_1.png.

③sin(2nπ+word/media/image21_1.png)=sinword/media/image21_1.png.

⑤sin［(2n+1)π-word/media/image21_1.png］=sin［2nπ+(π-word/media/image21_1.png)］=sin(π-word/media/image21_1.png)=sinword/media/image21_1.png.

答案：C

4.若cos(π+α)=word/media/image46_1.png,且α∈(-word/media/image15_1.png，0)，则tan(word/media/image47_1.png+α)的值为( )

A.word/media/image48_1.png B.word/media/image49_1.png C.word/media/image50_1.png D.word/media/image51_1.png

解析：cos(π+α)=-cosα=word/media/image46_1.png,∴cosα=word/media/image52_1.png.

又α∈(-word/media/image15_1.png,0),∴sinα=word/media/image53_1.png.

∴tan(word/media/image43_1.png+α)=-cotα=-word/media/image54_1.png=word/media/image49_1.png.

答案：B

5.设A、B、C是三角形的三个内角，下列关系恒等成立的是( )

A.cos(A+B)=cosC B.sin(A+B)=sinC

C.tan(A+B)=tanC D.sinword/media/image55_1.png=sinword/media/image56_1.png

解析：根据三角形的内角和及诱导公式判断.

答案：B

6.已知f(cosx)=2cos2x，则f(sin15°)等于( )

A.1 B.word/media/image44_1.png C.word/media/image5_1.png D.word/media/image57_1.png

解析：f(sin15°)=f［sin(90°-75°)］=f(cos75°)=2cos(2×75°)=2cos150°

=2cos(180°-30°)=-2cos30°=word/media/image58_1.png.

答案：C

7.sin2(word/media/image59_1.png-x)+sin2(word/media/image11_1.png+x)=_______________.

解析：观察出(word/media/image21_1.png-x)+(word/media/image11_1.png+x)=word/media/image15_1.png,再利用诱导公式化为sin2α+cos2α=1的形式求解.

答案：1

8.sin21°+sin22°+sin23°+…+sin289°=_______________.

解析：把给定式子利用诱导公式化为sin2α+cos2α=1的形式，再求和.

答案：word/media/image60_1.png

9.化简：word/media/image61_1.png+sin(-θ).

解：word/media/image62_1.png+sin(-θ)

=word/media/image63_1.png+sin(-θ)

=word/media/image64_1.png-sinθ

=word/media/image65_1.png-sinθ

=word/media/image66_1.png-sinθ

=1-sinθ.

10.已知cos(75°+α)=word/media/image67_1.png，其中α为第三象限角，求cos(105°-α)+sin(α-105°)的值.

解：cos(105°-α)=cos［180°-(75°+α)］=-cos(75°+α)= word/media/image68_1.png.

sin(α-105°)=-sin(105°-α)=-sin［180°-(75°+α)］=-sin(75°+α).

∵cos(75°+α)=word/media/image67_1.png＞0,又α为第三象限角，可知角75°+α为第四象限角，

则有sin(75°+α)=word/media/image69_1.png.

故cos(105°-α)+sin(α-105°)=word/media/image70_1.png.

快乐时光

生 气

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一个学生回答：“蜜蜂偷了花蜜，花儿就生气了啊！”

大家听了笑个不停.

那学生又说：“笑什么？要是鲜花不生气，哪来鲜花怒放呢？”

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